Question

I need to implement a method for non-linear interpolation between values, ease-in, ease-out, general easing curves as well as user defined curves.

I have a basic idea of how to do this – but I am not sure if it will be the most efficient solution. My idea is basically as follows:

Use a 2D cubic, quadratic or n-th order Bezier curve to control the interpolation. Traverse through the curve linearly to get the non-linear Y component, and use that to value to feed a simple linear interpolation method:

value = v1 + (v2 - v1) * t;

Where t is the non-linear Y component of the control curve.

This allows for custom, user defined methods for interpolation, but it comes at a cost, the cost of one non-linear interpolation is equal to:

1 + 2 * (n-1)

total interpolations, where n is the order, or number of control points of the control curve.

I am NO MATHEMATICIAN, this is the best I could come up with, so my question is if there is a better solution?

EDIT: I am probably not explaining it right, I am not a native speaker, so here is something hopefully everyone will understand:

Answer 1

From what I understand, your t is actually a family of functions f_i(u), where both u, and f_i(u) are between 0 and 1. If that is the case, it doesn’t get any better than what you’ve already proposed.

It looks like you are worried about evaluating these f_i(u) values during actual curve calculation. There is no avoiding the evaluation if you don’t want to pre-calculate. If performance is a big issue and you don’t need to be very precise, you can calculate tables of f_i(u_j) for as many u_j values as you want (say 100 or 1000 discrete points between 0 and 1) for each of your curves, and when you need a value between your sampling points, do a simple linear interpolation of the two cached values around your desired point.